Problem: What is the extraneous solution to these equations? $\dfrac{x^2 - 23}{x - 4} = \dfrac{13x - 59}{x - 4}$
Answer: Multiply both sides by $x - 4$ $ \dfrac{x^2 - 23}{x - 4} (x - 4) = \dfrac{13x - 59}{x - 4} (x - 4)$ $ x^2 - 23 = 13x - 59$ Subtract $13x - 59$ from both sides: $ x^2 - 23 - (13x - 59) = 13x - 59 - (13x - 59)$ $ x^2 - 23 - 13x + 59 = 0$ $ x^2 + 36 - 13x = 0$ Factor the expression: $ (x - 4)(x - 9) = 0$ Therefore $x = 4$ or $x = 9$ At $x = 4$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 4$, it is an extraneous solution.